The Long Line
Richard Koch
November 24, 2005
1
Introduction
Before this class began, I asked several topologists what I should cover in the first term.
Everyone told me the same thing: go as far as the classification of compact surfaces. Having
done my duty, I feel free to talk about more general results and give details about one of
my favorite examples.
We classified all compact connected 2dimensional manifolds. You may wonder about the
corresponding classification in other dimensions.
It is fairly easy to prove that the only
compact connected 1dimensional manifold is the circle
S
1
; the book sketches a proof of
this and I have nothing to add.
In dimension greater than or equal to four, it has been proved that a complete classification
is impossible (although there are many interesting theorems about such manifolds). The
idea of the proof is interesting: for each finite presentation of a group by generators and re
lations, one can construct a compact connected 4manifold with that group as fundamental
group. Logicians have proved that the word problem for finitely presented groups cannot
be solved. That is, if I describe a group
G
by giving a finite number of generators and a
finite number of relations, and I describe a second group
H
similarly, it is not possible to
find an algorithm which will determine in all cases whether
G
and
H
are isomorphic. A
complete classification of 4manifolds, however, would give such an algorithm.
As for the theory of compact connected three dimensional manifolds, this is a very ex
citing time to be alive if you are interested in that theory.
Three dimensional compact
manifolds have been studied intensively since the work of Poincare around 1900. In the
1970’s, Bill Thurston expanded this theory dramatically; roughly speaking, he showed
that every such manifold can be canonically cut into pieces of a special nature, and
then conjectured that each of these pieces can be given a particular kind of geometry.
The geometry dramatically decreases the possible topologies of that piece.
See for ex
ample,
http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html
for more
details.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Around 1980, a possible approach to proving the Thurston conjecture was developed by
R. S. Hamilton. Two faculty members at Oregon, Peng Lu and Jim Isenberg, are experts
in this theory. In 2002, G. Perelman announced a proof of the full Thurston conjectures
using Hamilton’s technique. Experts are verifying his proof now; the experts believe that
there is a good chance it is correct. Perelman’s proof will not complete the classification,
but it will push us astonishingly closer to that goal.
One special case of Perelman’s result would give all possible compact 3manifolds with
finite fundamental group.
In particular, it would show that
S
3
is the only compact 3
manifold with trivial fundamental group, so that every closed loop can be deformed to
a point.
Proving this special case is one of seven unsolved problems for which the Clay
Institute offers a prize of $1,000,000 for a solution.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 N.R.Wallach
 Topology, Order theory, Natural number, Open set, Topological space, Ordinal number

Click to edit the document details